Friends, I have a project already running, but I need help to improve it and would like to hear your opinions.

I have several boxes that I need to maintain the control over the inside temperature, because I have some alive samples inside.

I know that the desired inside temperature is always higher than room temperature, so I'll never have to cool the boxes, but only heat them.

The resistance used to heat each them are two incandescent bulbs in series (microwave bulbs). I used a relay to turn on the lights when the temperature drops below the minimum, and turn them off when the temperature rises above the maximum.

This was good, because I had an acceptable temperature range. But my new samples need an exact temperature, with a minimal tolerance. And turn on/off the lamps in short intervals of time is not good enough.

Using a manual potentiometer to control the intensity of the lamps is not good, because sometimes I need to stay many hours away from the boxes, and the ambience temperature reduces at night, and rises during the day.

Someone who understands of electronics know how to inform what kind of cheap component can I use to control the intensity of the lamps through my arduinos?

I need something like a relay, instead of on / off only, would have several intermediate points.

I'm talking about controlling the intensity of lamps that are like 110v-220v AC, 50-60 Hz.

With a fairly small amount of parts you should be able to set up PWM modulation of your light globes, effectively giving you a wide range of temperatures you could set them at. Warning, this project involves mains electricity!

For more control you may want to combine that with the PID library:

http://playground.arduino.cc/Code/PIDLibrary

I haven't used that either, but I gather it should adjust the output (in your case, the amount of PWM) to correspond to your set point (desired temperature) based on the reading of your thermometer.

Please post technical questions on the forum, not by personal message. Thanks!